

To obtain the “matter" contribution in eq.(REF ) for the conjugate momentum \(\pi ^{ij}_{x}\) , note that the metric appears in \(\tilde{H}_{\perp x}\)  through the density \(g_{x}^{1/2}\)  and the inverse metric \(g^{ij}_{x}\) . Variations of these quantities with respect to the metric \(g_{ijx}\)  are given by [1]}
\(\delta g_{x} = g \,g^{ij}_{x} \, \delta g_{ijx}\quad \text{and}\quad \delta g^{ij}_{x} = g^{ik}g^{jl}\delta g_{kl}~.\) 
